Internal Model Control Design for Nonlinear Systems Based on Inverse Dynamic Takagi–Sugeno Fuzzy Model

Abstract

In recent years, applications of inverse model-based control techniques have experienced significant growth in popularity and have been widely used in engineering applications, mainly in nonlinear control system design problems. In this study, a novel fuzzy internal model control (IMC) structure is presented for single-input-single-output (SISO) nonlinear systems. The proposed structure uses the forward and inverse dynamic Takagi–Sugeno (D-TS) fuzzy models of the nonlinear system within the IMC framework for the first time in literature. The proposed fuzzy IMC is obtained in a two-step procedure. A SISO nonlinear system is first approximated using a D-TS fuzzy system, of which the rule consequents are linearized subsystems derived from the nonlinear system. A novel approach is used to achieve the exact inversion of the SISO D-TS fuzzy model, which is then utilized as a control element within the IMC framework. In this way, the control design problem is simplified to the inversion problem of the SISO D-TS fuzzy system. The provided simulation examples illustrate the efficacy of the proposed control method. It is observed that SISO nonlinear systems effectively track the desired output trajectories and exhibit significant disturbance rejection performance by using the proposed control approach. Additionally, the results are compared with those of the proportional-integral-derivative control (PID) method, and it is shown that the proposed method exhibits better performance than the classical PID controller.

1. Introduction

Approaches to controlling nonlinear dynamical systems based on conventional methods, such as feedback linearization [1], Lyapunov function-based control [2], and nonlinear model predictive control [3], provide effective control solutions for a certain type of nonlinear system. However, the control performance of these methods depends on the availability of the exact model of the nonlinear dynamical systems. Moreover, the variations in model parameters affect the robustness of the controller. These limitations can be minimized with the integration of a D-TS fuzzy model within the IMC structure [4].

The concept of the IMC was first presented by Morari et al. in their study, [5]. Recently, conventional IMC has undergone improvements utilizing soft computing methods, including the application of Artificial Neural Networks (ANNs) and fuzzy inference systems to enhance its performance [6,7]. For example, in [8,9], an ANN approach is adopted for the synthesis of the IMC structure, where feedforward NNs are trained with input-output data during model identification of the nonlinear system. The obtained model is then used as the internal model of the IMC and the inverse model is derived by simply swapping the training data set. However, there are some limitations related to the above techniques, such as convergence conditions during the training of the NNs, the size of the training data, and the approximation error.

Alternative approaches to implement the IMC scheme utilize the fuzzy internal model of the nonlinear dynamical system. From the literature, TS fuzzy systems have the ability to approximate any nonlinear system having continuous functions with a high degree of accuracy [10,11,12]. The application of TS fuzzy models is fundamental in the development of model-based control techniques [13,14,15], particularly when used in the trajectory tracking problem in nonlinear systems, as demonstrated in [16,17]. The IMC scheme based on the TS fuzzy systems demonstrates good robustness performance. They have the capability to demonstrate robustness to parameter variations, external disturbances, and measurement noise. A key feature of the fuzzy IMC approach is that the control element can be obtained by using the inverted TS fuzzy forward model for any given stable system, as demonstrated in [18]. Here, a D-TS fuzzy model of the plant is used as a reference model; the controller design is then approximated by inverting the D-TS fuzzy model. In the literature, there are several studies related to the inversion of the TS fuzzy systems [19,20,21,22,23,24,25,26,27,28]. However, there are a few limitations related to the derivation of the inverse fuzzy models, including a monotonicity requirement for the fuzzy rule base, or a decomposability requirement for the fuzzy model. Additionally, these approaches give inverse solutions, particularly for fuzzy systems with linear or singleton consequents.

Fuzzy systems having linear or singleton consequents provide only a static mapping between their inputs and outputs. In these fuzzy models, the inputs are generally chosen from the preceding values of the system output and the input. However, D-TS fuzzy systems with linearized subsystems of the given nonlinear system as rule consequents provide a more compact representation of the nonlinear system and also simplify the controller design [29]. Furthermore, D-TS fuzzy systems allow stability analyses using conventional methods in controller design since they are directly derived from the nonlinear system analytical model. Due to these advantages, D-TS fuzzy models are commonly used in control design applications [30,31,32,33,34]. However, the inverse of D-TS fuzzy models has not yet been applied in IMC applications in the literature due to the lack of an available inversion method. A significant advantage of the IMC approach lies in its ability to simplify the design of nonlinear controllers [35]. The IMC design procedure, including the robust filter, ideally leads to PID-type equivalent controller [36,37].

In this study, a novel fuzzy IMC structure based on an exact inverse D-TS fuzzy model is presented for SISO nonlinear systems. The control design procedure is simplified by inverting the TS fuzzy model of the nonlinear system. This allows the control element to be easily obtained for any stable system, enabling the achievement of the desired trajectory-tracking responses. Such flexibility enhances the effectiveness of the fuzzy IMC approach. The proposed design procedure entails obtaining a model of the SISO nonlinear system using a D-TS fuzzy system. The exact inverse definitions are then derived for all linear subsystems in the rule consequents of this fuzzy system. The exact inverse D-TS fuzzy model is constructed by replacing the rule consequents with inverse definitions of the corresponding linear state-space models, provided that the rule antecedents remain the same. The derived inverse D-TS fuzzy model is then used as the main control element in the proposed IMC structure to achieve robust and precise trajectory tracking for SISO nonlinear systems. The validity of the control scheme is demonstrated through simulation studies.

This paper is structured as follows. The construction of a D-TS fuzzy model for a general SISO nonlinear system is provided in Section 2. Section 3 presents the proposed IMC structure based on the exact inverse D-TS fuzzy model of the SISO D-TS fuzzy system. Section 4 provides simulation examples. Conclusions are outlined in Section 5.

2. D-TS Fuzzy Model Structure

A D-TS fuzzy model for SISO affine nonlinear systems is presented in this section. Consider a general SISO affine nonlinear system whose dynamic equation is given by

$$\begin{array}{cc}\hfill x_{k+1}& =f\left(x_k\right)+g\left(x_k\right)u_k\hfill \\ \hfill y_k& =h\left(x_k\right)\hfill \end{array}$$

(1)

Here, $x_k\in {\Re }^n$, $u_k\in \Re$ and $y_k\in \Re$ are system states, input and output from the nonlinear system, respectively. $f\left(x_k\right),g\left(x_k\right)$ and $h\left(x_k\right)$ are considered continuous functions that depict the nonlinear dynamics of the system.

A D-TS fuzzy model is then constructed to approximate the dynamical behavior of the nonlinear system within a given interval. The rule consequents of the D-TS fuzzy model consist of linear state-space equations obtained by linearizing the nonlinear system about specific operating points [38]. In this way, the mapping between the input and output of the nonlinear system is reduced to a combination of linearized subsystems with their corresponding membership degree of the fuzzy rules.

The fuzzy IF-THEN rules have the form as shown below.

$$\begin{array}{c}\hfill {\mathbf{R}}^{i_1,i_2,\cdots ,i_n}:\mathbf{IF}{x}_{1_k}is{M}_1^{i_1}\mathbf{AND}{x}_{2_k}is{M}_2^{i_2}\\ \hfill \mathbf{AND}\cdots \mathbf{AND}{x}_{n_k}is{M}_n^{i_n}\mathbf{THEN}\\ \hfill x_{k+1}^{i_1,i_2,\cdots ,i_n}=A^{i_1,i_2,\cdots ,i_n}{x}_k+B^{i_1,i_2,\cdots ,i_n}{u}_k\\ \hfill y_k^{i_1,i_2,\cdots ,i_n}=C^{i_1,i_2,\cdots ,i_n}{x}_k\end{array}$$

(2)

Here, $i_k\in I_k\in \{i_1,i_2,\cdots ,N_k\}$ and $k=1,2,\cdots ,n$. The complete fuzzy rule base is composed of $N={\prod }_{k=1}^n{N}_k$ rules and the index set is $I_1\times I_2\times ,\cdots ,\times I_n$ [19]. In the rule consequents, $A\in {\Re }^{n\times n}$, $B\in {\Re }^{n\times 1}$ and $C\in {\Re }^{1\times n}$ denote the state, input and output matrices, respectively. Where, $x_k$ constitutes the premise variables, obtained from the nonlinear equation for each linear subsystem. $M_j^{i_j}$ is the fuzzy set, and ${\mu }_{M^{i_j}}\left(x_{j_k}\right)$ denotes the grade of membership of $x_{j_k}$. Triangular membership functions are utilized with strong partitions to represent fuzzy sets within the various universes of discourse (Figure 1).

The boundaries are directly determined with respect to the physical limits and working conditions of the system variables. If these boundaries are not properly determined, the dynamic behavior of the nonlinear system beyond these boundaries is represented with lower accuracy by the D-TS fuzzy model, leading to a degradation in control performance.

Finally, the output of the D-TS fuzzy model is obtained by applying the fuzzy-mean defuzzification [39] method as follows.

$$x_{k+1}=\frac{{\sum }_{(i_1,i_2,\cdots ,i_n)}{\omega }^{i_1,i_2,\cdots ,i_n}{x_{k+1}}^{i_1,i_2,\cdots ,i_n}}{{\sum }_{(i_1,i_2,\cdots ,i_n)}{\omega }^{i_1,i_2,\cdots ,i_n}}$$

(3)

$$y_k=\frac{{\sum }_{(i_1,i_2,\cdots ,i_n)}{\omega }^{i_1,i_2,\cdots ,i_n}{y_k}^{i_1,i_2,\cdots ,i_n}}{{\sum }_{(i_1,i_2,\cdots ,i_n)}{\omega }^{i_1,i_2,\cdots ,i_n}}$$

(4)

${\omega }^{i_1,i_2,\cdots ,i_n}$ represents the firing strength of the antecedent part of each rule and is derived as follows:

$${\omega }^{i_1,i_2,\cdots ,i_n}=\prod _{j=1}^n{\mu }_{M_j^{i_j}}\left(x_{j_k}\right)$$

(5)

Considering that the triangular fuzzy membership function that forms strong partitioning defines the fuzzy sets $M_j^{i_j}$, hence in Equations (4) and (5) the denominator becomes ${\sum }_{(i_1,i_2,\cdots ,i_n)}{\omega }^{i_1,i_2,\cdots ,i_n}=1$, the output of the D-TS fuzzy model is as shown below.

$$x_{k+1}=\sum _{(i_1,i_2,\cdots ,i_n)}{\omega }^{i_1,i_2,\cdots ,i_n}{x}_{k+1}^{i_1,i_2,\cdots ,i_n}$$

(6)

$$y_k=\sum _{(i_1,i_2,\cdots ,i_n)}{\omega }^{i_1,i_2,\cdots ,i_n}{y}_k^{i_1,i_2,\cdots ,i_n}$$

(7)

3. Fuzzy IMC System Design Based on the Exact Inverse Dynamic TS Fuzzy Model

The proposed fuzzy IMC structure is shown in Figure 2. In this structure, an exact inverse model of the D-TS fuzzy model is used as the main control element. The nonlinear system is initially modeled using a D-TS fuzzy system within a specified range of operation. The inversion of the D-TS fuzzy model is then obtained using the proposed approach. The inversion approach used in this study offers a systematic technique for obtaining an exact inverse D-TS fuzzy model. A fundamental invertibility property of the relative degree of a system is involved in deriving the inverse model of the D-TS fuzzy system. This property is commonly used to signify the invertibility of a nonlinear system, especially in feedback linearization control design [40]. These techniques are applicable to input state linearizable (or minimum phase) systems that are completely state-observable. The derivation of relative degree as formulated by Isidori [40] for SISO time-invariant nonlinear systems with affine control input as given in Equation (2) is having a relative degree $r\in \mathbb{N}$, $1\ge r\le n$ as long as there is an open neighborhood $\mathcal{X}$ of x, such that

$$\begin{array}{c}\forall x\in \mathcal{X}\forall i\in \{1,2,\dots ,r-1\}\\ L_g{L}_f^{i-1}h\left(x\right)=0,L_g{L}_f^{r-1}h\left(x\right)\ne 0\end{array}$$

(8)

$$y^{\left(r\right)}=L_f^{r}h\left(x\right)+L_g{L}_f^{r-1}h\left(x\right)u$$

(9)

where $L_{f}h\stackrel{\Delta }{=}{\displaystyle \frac{\partial h}{\partial x}}f$ denotes the partial derivative of h along f.

Consider a general class of SISO linear systems of the form

$$\left.\begin{array}{c}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),x\left(0\right)=x_0\in {\Re }^n,\hfill \\ y\left(t\right)=Cx\left(t\right)\in \Re ,\hfill \end{array}\right\}$$

(10)

Thus, according to [41] the following is also true for the relative degree of Equation (10),

$$y^{\left(r\right)}\left(t\right)=C{A}^{r}x\left(t\right)+C{A}^{r-1}Bu\left(t\right)$$

(11)

and

$$\begin{array}{c}\forall k\in \{0,\dots ,r-2\}\\ L_g{L}_f^{k}h\left(x\left(t\right)\right)=C{A}^{k}B=0,L_g{L}_f^{r-1}h\left(x\left(t\right)\right)=C{A}^{r-1}B\ne 0.\end{array}$$

(12)

In the discrete-time domain Equation (11) becomes,

$$y_{k+r}=C{A}^r{x}_k+C{A}^{r-1}B{u}_k$$

(13)

and the inverse of the system is given by,

$$u_k={\left(C{A}^{r-1}B\right)}^{-1}(y_{k+r}-C{A}^r{x}_k)$$

(14)

The above formulation indicates that the existence of a well-defined relative degree implies that the system is invertible and an inverse of the system can be defined using Equation (14).

Hence, the concept of determining the relative degree can be extended to the consequent terms of the SISO D-TSF model in the discrete-time domain. The linearized sub-systems of the D-TSF model for all the rules, $I\in (i_1,i_2,\cdots ,i_n)$ can be represented by the following expressions.

$$x_{k+1}=A^{i_1,i_2,\cdots ,i_n}{x}_k+B^{i_1,i_2,\cdots ,i_n}{u}_k$$

(15)

$$y_k=C^{i_1,i_2,\cdots ,i_n}{x}_k$$

(16)

Hence, using Equation (11) the following expression is obtained using the linearized subsystems of the D-TSF model,

$$\begin{array}{cc}\hfill y_{k+r}& =C^{i_1,i_2,\cdots ,i_n}(A^{i_1,i_2,\cdots ,i_n}{)}^r{x}_k+C^{i_1,i_2,\cdots ,i_n}{\left(A^{i_1,i_2,\cdots ,i_n}\right)}^{r-1}{B}^{i_1,i_2,\cdots ,i_n}{u}_k\hfill \end{array}$$

(17)

The coefficients in Equation (17) can be represented as follows

$$\begin{array}{cc}\hfill {\tilde{D}}^{i_1,i_2,\cdots ,i_n}& \stackrel{\Delta }{=}{C}^{i_1,i_2,\cdots ,i_n}{\left(A^{i_1,i_2,\cdots ,i_n}\right)}^{r-1}{B}^{i_1,i_2,\cdots ,i_n}\hfill \\ \hfill {\tilde{C}}^{i_1,i_2,\cdots ,i_n}& \stackrel{\Delta }{=}{C}^{i_1,i_2,\cdots ,i_n}(A^{i_1,i_2,\cdots ,i_n}{)}^r\hfill \end{array}$$

(18)

Equation (17) is then rewritten as shown below,

$$y_{k+r}={\tilde{C}}^{i_1,i_2,\cdots ,i_n}{x}_k+{\tilde{D}}^{i_1,i_2,\cdots ,i_n}{u_k}^{i_1,i_2,\cdots ,i_n}$$

(19)

Therefore, the inverse of the linear system can be determined directly from Equation (19) as follows.

$${u_k}^{i_1,i_2,\cdots ,i_n}={\left({\tilde{D}}^{i_1,i_2,\cdots ,i_n}\right)}^{-1}(y_{k+r}-{\tilde{C}}^{i_1,i_2,\cdots ,i_n}{x}_k)$$

(20)

The proposed approach is suitable for minimum-phase systems, ensuring stable inversion. Hence, using Equation (20), the consequent parts of the rules of the TS fuzzy inverse model can easily be derived [28]. Suppose the state vector is measurable as $x_k\in {\Re }^n$ at any given discrete time k with triangular fuzzy membership function defining the fuzzy sets for the antecedent variables, $M_j^{i_j}$. In this case, the exact inverse D-TS fuzzy model of the forward D-TS fuzzy model given in Equation (2) is formulated as follows [28]

$$\begin{array}{c}\hfill {\mathbf{R}}^{i_1,i_2,\cdots ,i_n}:\mathbf{IF}{x}_{1_k}is{M}_1^{i_1}\mathbf{AND}{x}_{2_k}is{M}_2^{i_2}\\ \hfill \mathbf{AND}\cdots \mathbf{AND}{x}_{n_k}is{M}_n^{i_n}\mathbf{THEN}\\ \hfill u_k^{i_1,i_2,\cdots ,i_n}={\left({\tilde{D}}^{i_1,i_2,\cdots ,i_n}\right)}^{-1}(y_{k+r}-{\tilde{C}}^{i_1,i_2,\cdots ,i_n}{x}_k)\end{array}$$

(21)

The output definition of the inverse D-TS fuzzy system is also obtained using fuzzy-mean defuzzification, as given in Equation (22)

$$u_k=\sum _{(i_1,i_2,\cdots ,i_n)\in I}{\omega }^{i_1,i_2,\cdots ,i_n}{u}_k^{i_1,i_2,\cdots ,i_n}$$

(22)

After constructing the inverse D-TS fuzzy model, it is then utilized as the main control element in the proposed IMC structure.

The proposed inversion technique is constrained to input state linearizable (or minimum phase) SISO systems which are fully state observable. This requirement ensures stable and exact inversion, as detailed in [42]. Given that the D-TS fuzzy model represents the nonlinear system perfectly, the exact inverse model provides perfect tracking in the absence of disturbance [18]. However, model mismatches, uncertainties, and disturbances are inevitable in practical applications. Therefore, to provide convergence, the effects of model mismatches, uncertainties, and also disturbances are eliminated by using an error feedback signal (between the system output and the model output) in the IMC structure [42]. Therefore, the convergence is guaranteed as stated in [18,42].

4. Simulation Examples

The effectiveness of the proposed control scheme using an inverse D-TS fuzzy system within the IMC framework is demonstrated in this section using two simulation examples. The results are compared with those of the PID control approach, which is the most widely used control method in practical applications. In the comparisons, the Integral of Squared Error (ISE) and Integral of Absolute Error (IAE) performance metrics are utilized and defined as follows:

• ISE $={\int }_0^{\infty }{e}^2\left(t\right)dt$
• IAE $={\int }_0^{\infty }\left|e\left(t\right)\right|dt$

4.1. Example A

In this example, the trajectory tracking problem of a nonlinear mass-spring mechanical system [43] is considered as depicted in Figure 3. The dynamic equation is derived as follows:

$$m\ddot{x}\left(t\right)+c\dot{x}\left(t\right)+(k+k{a}^{2}x{\left(t\right)}^2)x\left(t\right)=u\left(t\right)$$

(23)

Here, $c=0.4$ Nms/rad is the damping coefficient, $m=1$ kg is the mass, $k=1.1$ and $a^2=0.9$ are the coefficients of the spring. x is the displacement of the mass and u is the control input.

Considering $x=x_1$ and $\dot{x}=x_2$, Equation (23) can be represented in the state space form as shown below.

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -k-k{a}^2{x_1}^2& -c\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u\left(t\right)$$

(24)

Using the sector nonlinearity approach [38], let the nonlinear term ${x_1}^2=z_1$. The fuzzy set interval of the antecedent variable is given by $z_1\in \left[\overline{d}\overline{D}\right]$, where $\overline{d}=0$ and $\overline{D}=1.2$ for this example. The membership functions which provide a strong triangular partitioning are computed as follows:

$$M_1^1={\displaystyle \frac{z_1-\overline{d}}{\overline{D}-\overline{d}}},M_1^2=1-M_1^1$$

(25)

The D-TS fuzzy model of the nonlinear system is derived as shown below.

$$\begin{array}{c}\hfill {\mathbf{R}}^i:\mathbf{IF}{z}_{1}is{M}_1^i\mathbf{THEN}{x}_{k+1}^i=A^i{x}_k+B^i{u}_k,y_k^i=C^{i}x,i\in I=\{1,2\}\end{array}$$

The linear subsystem matrices obtained using the sector nonlinearity method on the nonlinear system are then discretized using a sampling time of $0.01$ s.

$$A^1=\left[\begin{array}{cc}0.9999& 0.00998\\ -0.02283& 0.9959\end{array}\right],A^2=\left[\begin{array}{cc}1.0000& 0.00998\\ -0.01098& 0.9960\end{array}\right]$$

$$B^1=B^2=\left[\begin{array}{c}4.993\times {10}^{-5}\\ 0.00998\end{array}\right],C^1=C^2=\left[\begin{array}{cc}1& 0\end{array}\right]$$

The linear subsystems in the rule consequents are fully controllable and fully observable. They are also minimum-phase systems and have a relative degree, $r=2$. Thus, the inverse D-TS fuzzy model is constructed as follows:

$$\begin{array}{c}\hfill {\mathbf{R}}^i:\mathbf{IF}{z}_{1}is{M}_1^i\mathbf{THEN}{u}_k^i={\left({\tilde{D}}^i\right)}^{-1}(y_{k+2}-{\tilde{C}}^i{x}_k),i\in I=\{1,2\}\end{array}$$

The values of the matrices, ${\tilde{D}}^i=C^i{A}^i{B}^i$ and ${\tilde{C}}^i=C^i{\left(A^i\right)}^2$, in rule consequents are as follows:

$${\tilde{C}}^1=[0.9996,0.0199],{\tilde{C}}^2=[1.0000,0.0199],{\tilde{D}}^1={\tilde{D}}^2=1.4953\times {10}^{-4}.$$

The initial conditions, $x_o=\left[0.10\right]$ are used in the simulation. The following reference input signal, $y_{ref}\left(t\right)=sin\left(t\right)$ is used for trajectory tracking, and the step function signal $d\left(t\right)$ is used as a disturbance.

$$d\left(t\right)=\left\{\begin{array}{cc}0\hfill & t\le 4\hfill \\ 0.1\hfill & t>4\hfill \end{array}\right.$$

(26)

In the proposed fuzzy IMC structure, the robustness filter is chosen as

$$F\left(z\right)=\frac{0.1535}{z-0.8465}$$

(27)

To show the effectiveness of the proposed method, the performance of the proposed method is compared with those of a PID controller. The PID parameters are as follows, $K_P=120$, $K_I=10$ and $K_D=5$. Figure 4 and Figure 5 provide results of the simulation.

Integral square error (ISE) and integral absolute error (IAE) values are used as performance metrics in the comparison. Based on these performance metrics for the fuzzy IMC system response, the ISE = 0.001896, and the IAE = 0.04464, whereas for PID, the ISE = 0.002861, and the IAE = 0.08796. It can be shown from the results that a more accurate trajectory tracking of the reference signal is achieved and the disturbance is suppressed in a short period using the proposed fuzzy IMC approach. Additionally, from Figure 5 the provided control signal $u_k$ is quite applicable. These results validate the effective trajectory tracking and robustness performance of the proposed fuzzy IMC scheme.

4.2. Example B

Consider a voltage tracking problem of the PWM Buck Converter given Figure 6. The dynamic behavior of the nonlinear system is represented as follows [43]:

$$\left[\begin{array}{c}{\dot{i}}_L\\ {\dot{v}}_C\end{array}\right]=\left[\begin{array}{cc}0& -{\displaystyle \frac{1}{L}}\\ \\ {\displaystyle \frac{1}{C}}& -{\displaystyle \frac{1}{RC}}\end{array}\right]\left[\begin{array}{c}{i}_L\\ v_C\end{array}\right]+\left[\begin{array}{c}-{\displaystyle \frac{1}{L}}(R_M{i}_L-V_{in}-V_D)\\ 0\end{array}\right]u\left(t\right)+\left[\begin{array}{c}-{\displaystyle \frac{V_D}{L}}\\ 0\end{array}\right]$$

(28)

Here, $v_c$ denotes the output voltage from the converter and $u\left(t\right)$ is the control input (duty ratio). The values of the circuit parameters are given as $L=98.58$ mH, $C=0.20250$ mF, $R=6\mathsf{\Omega }$, $R_M=0.27\mathsf{\Omega }$, $V_D=0.820$ V and $V_{in}=30.0$ V.

Consider $i_L=x_1$ and $v_c=x_2$. $x_1$ is chosen as the antecedent variable and the corresponding universe of discourse is chosen as $x_1\in \left[\overline{d}\overline{D}\right]$ where $\overline{d}=0$, $\overline{D}=4$. The membership functions are computed as follows:

$$M_1^1={\displaystyle \frac{x_1-\overline{d}}{\overline{D}-\overline{d}}},M_1^2=1-M_1^1$$

(29)

The D-TS fuzzy structure of the buck converter is derived as shown below

$$\begin{array}{c}\hfill {\mathbf{R}}^i:\mathbf{IF}{x}_{1_k}is{M}_1^i\mathbf{THEN}{x}_{k+1}^i=A^i{x}_k+B^i{u}_k+W^i,y_k^i=C^{i}x,i\in I=\{1,2\}\end{array}$$

The system given in Equation (28) is linearized using the sector nonlinearity approach [38] and discretized using a sampling time of $0.001$ s. The system matrices in the rule consequents are obtained as follows:

$$A^1=A^2=\left[\begin{array}{cc}0.9807& -0.006856\\ 3.338& 0.4244\end{array}\right],C^1=C^2=\left[\begin{array}{cc}0& 1\end{array}\right]$$

$$B^1=\left[\begin{array}{c}0.2996\\ 0.5742\end{array}\right],B^2=\left[\begin{array}{c}0.3105\\ 0.5951\end{array}\right],W^1=W^2=\left[\begin{array}{c}-0.008261\\ -0.01583\end{array}\right]$$

For the given TS fuzzy model, all linear subsystems in the rule consequents are fully controllable and fully observable, and minimum phase systems have a relative degree, r = 1. Therefore, the inverse D-TS fuzzy model is obtained as shown.

$$\begin{array}{c}\hfill {\mathbf{R}}^i:\mathbf{IF}{x}_{1_k}is{M}_1^i\mathbf{THEN}{u}_k^i={\left({\tilde{D}}^i\right)}^{-1}(y_{k+1}-{\tilde{C}}^i{x}_k-C^i{W}^i),i\in I=\{1,2\}\end{array}$$

The numerical values of the matrices, ${\tilde{D}}^i=C^i{B}^i,{\tilde{C}}^i=C^i{A}^i$ and $C^i{W}^i$ in the rule consequents are as follows

$${\tilde{C}}^1={\tilde{C}}^2=[3.338,0.4244],{\tilde{D}}^1=0.5742,{\tilde{D}}^2=0.5951,C^1{W}^1=C^2{W}^2=-0.01583.$$

The initial condition is considered as $x_o=\left[0.10.1\right]$ in the simulation. The applied reference input signal $y_{ref}\left(t\right)$ and the disturbance signal $d\left(t\right)$ are given below.

$$y_{ref}\left(t\right)=\left\{\begin{array}{cc}0\hfill & t\le 0\hfill \\ 10\hfill & 0<t\le 0.35\hfill \\ 2\hfill & t>0.35\hfill \end{array}\right.$$

(30)

$$d\left(t\right)=\left\{\begin{array}{cc}0\hfill & t\le 0.2\hfill \\ 1\hfill & t>0.2\hfill \end{array}\right.$$

(31)

In this example, the robustness filter is chosen as

$$F\left(z\right)=\frac{0.2835}{z-0.7165}$$

(32)

For the performance comparison, a PI controller with gain values of $K_P=0.07$ and $K_I=4$ is used. Figure 7 illustrates the system responses while Figure 8 shows the control signals. Performance metrics are obtained as IAE = 0.1527 and ISE = 0.7898 for fuzzy IMC whereas for the PI control approach, IAE = 0.2014 and ISE = 0.8609. As seen in Figure 7, by using the proposed method the output voltage of the buck converter not only tracks the reference voltage signal accurately but also exhibits better robustness performance against disturbance compared to the PI controller. Furthermore, Figure 8 clearly shows that the proposed fuzzy IMC structure provides an applicable control signal.

5. Conclusions

In this research study, a novel fuzzy IMC structure is presented for SISO nonlinear systems. The proposed structure uses the forward and inverse D-TS fuzzy models of the nonlinear system within the IMC framework. In this method, the exact inverse D-TS fuzzy model is used as the main controller. The effectiveness and robustness of the designed control system are validated by the simulations carried out on two SISO nonlinear dynamic systems.

The design procedure for the proposed fuzzy IMC system is straightforward. A forward D-TS fuzzy model for a SISO nonlinear system can be easily derived from the system analytical model by using linearization approaches. After the derivation of the forward model, its exact inverse D-TS fuzzy model is derived by inverting the linear systems in the rule consequents using simple formulations. The inverse model is then utilized as the control element in the proposed fuzzy IMC structure. Thus, it can be assumed that the controller design problem is reduced to the inversion problem of linear systems in the rule consequents.

In this research, it is assumed that the system under investigation is SISO nonlinear dynamical system. Hence, future research studies may focus on extending the proposed fuzzy IMC to include multi-input-multi-output time-delayed nonlinear systems with uncertainties.